Anisotropic Pauli spin blockade in hole quantum dots
Matthias Brauns,1, ∗ Joost Ridderbos,1 Ang Li,2 Erik P. A. M.
Bakkers,2, 3 Wilfred G. van der Wiel,1 and Floris A. Zwanenburg1
arXiv:1608.00111v1 [cond-mat.mes-hall] 30 Jul 2016
1
NanoElectronics Group, MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2
Department of Applied Physics, Eindhoven University of Technology,
Postbox 513, 5600 MB Eindhoven, The Netherlands
3
QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
(Dated: August 2, 2016)
We present measurements on gate-defined double quantum dots in Ge-Si core-shell nanowires,
which we tune to a regime with visible shell filling in both dots. We observe a Pauli spin blockade
and can assign the measured leakage current at low magnetic fields to spin-flip cotunneling, for which
we measure a strong anisotropy related to an anisotropic g factor. At higher magnetic fields we see
signatures for leakage current caused by spin-orbit coupling between (1,1)-singlet and (2,0)-triplet
states. Taking into account these anisotropic spin-flip mechanisms, we can choose the magnetic field
direction with the longest spin lifetime for improved spin-orbit qubits.
INTRODUCTION
For quantum computation [1–3], increasing research
efforts have focused in recent years on C, Si, and Ge [4–
6] because these materials can be purified to only consist of isotopes with zero nuclear spin [7, 8] and thus
exhibit exceptionally long spin lifetimes [9, 10]. The onedimensional character of Ge-Si core-shell nanowires leads
to unique electronic properties in the valence band, where
heavy and light hole states are mixed [11–13]. The band
mixing gives rise to an enhanced Rashba-type spin-orbit
interaction (SOI) [13], leading to strongly anisotropic and
electric-field dependent g factors [14, 15]. This makes
quantum dots in Ge-Si core-shell nanowires promising
candidates for robust spin-orbit qubits.
Crucial steps towards high-fidelity qubits are spin
readout via a Pauli spin blockade (PSB) [16] and understanding the spin relaxation mechanisms. For Ge-Si
core-shell nanowires, there have been reports on charge
sensing [17] spin coherence [18], and spin relaxation [19].
The authors of the latter performed their experiments
along a single magnetic field axis and concluded additional work was needed to pinpoint the spin relaxation
mechanism.
In this Rapid Communication, we define a hole double quantum dot in a Ge-Si core-shell nanowire by means
of gates. We observe shell filling and a Pauli spin blockade. The measured leakage currents strongly depend on
both the magnitude and direction of the magnetic field
and are assigned to spin-flip cotunneling processes for
low magnetic fields. We find signatures of SOI-induced
leakage current at higher magnetic fields up to 1 T.
SHELL FILLING AND PAULI SPIN BLOCKADE
Our device in Fig. 1(a) consists of a p ++ -doped Si substrate covered with 200 nm SiO2 , on which six bottom
gates with 100 nm pitch are patterned with electronbeam lithography (EBL). The gates are buried by 10 nm
Al2 O3 grown with atomic layer deposition at 100◦ C. A
single nanowire with a Si shell thickness of ∼2.5 nm and
a defect-free Ge core with a radius of ∼8 nm [20], in
preparation) is deterministically placed on top of the gate
structure with a micromanipulator and then contacted
with ohmic contacts made of 0.5/50 nm Ti/Pd. Based
on transmission electron microscopy studies of similar
nanowires the wire axis is most likely pointed along the
<110> crystal axis. A source-drain voltage VSD is applied between the source and ground, and the current
I is measured at the drain contact. All measurements
are performed using direct current (dc) electronic equipment in a dilution refrigerator with a base temperature
of 8 mK.
In Fig. 1(b) we plot I versus the voltage Vg3 on gate
g3 and the voltage Vg5 on g5. We see a highly regular
pattern of bias triangles [21], indicating the formation of
a double quantum dot above gates g3 (“left dot”) and
g5 (“right dot”). The 16 bias triangle pairs all have very
sharp edges, and the absence of current along the honeycomb edges indicates a double quantum dot weakly
coupled to the reservoirs [21]. We introduce (m, n) as
the effective charge occupation numbers m and n of the
left and right dot, respectively.
Nine honeycomb cells are visible in Fig. 1(b). In each
column from right to left a hole is added to the left dot,
while in each row from top to bottom a hole is added to
the right dot. The addition energy Eadd for each added
hole can be extracted from Fig. 1(b) by measuring the
distance between the triple points that are connected by
the dashed lines and converting this distance graphically
from the gate voltage into energy using the bias triangle size as a scale. The addition energy of the left dot
is Eadd = 9.8 ± 0.1 meV for the middle and left column, and Eadd = 10.3 ± 0.1 meV for the right column.
For the right dot Eadd = 9.5 ± 0.1 meV in the top and
2
g1
g2 g3 g4 g5
2585 mV
3195 mV
g6
QD1
wire
S
2210 mV
D
a)
|I| (pA) 0
350
QD2
g1 g2 g3 g4 g5 g6
0
4
8
I (pA)
b)
Vg5 (mV)
a)
342
VSD = +2 mV
(2,0)
Vg5 (mV)
(1,-1)
(0,-1)
(2,0)
(1,0)
(0,0)
µ 1(0,0)
360
µ 1(1,0)
Vg5 (mV)
µ2 (0,0)
(2,1)
(1,1)
460
(1,2)
500
Vg3 (mV)
B
PS
(2,0)
(0,2)
508
516
VSD = -2 mV
327
366
(2,0)
358
(1,1)
486
494
362 (2,0)
VSD = -2 mV
(1,1)
(1,1)
(1,1)
0 ε (meV) 1
0
blocked
0 ε (meV) 1
0
nonblocked
0 ε (meV) 1
FIG. 2. Zoom of the bias triangle pairs exhibiting PSB: (a)
and (b) with the same barrier gate voltages as in Fig. 1, and
(c) at Vg2 = 2570 mV, Vg4 = 2210 mV, and Vg6 = 3160 mV.
Line cuts (lowest panels) taken along the dotted lines for the
VSD direction with (blue) and without PSB (green).
(0,1)
320
(2,2)
486
VSD = -2 mV
321
B
PS
20
VSD = +2 mV
(0,2)
340
354 PSB
319
482V (mV)490
504V (mV) 512
474 V (mV) 482
g3
g3
g3
15
5
PSB
PSB
PSB
5
0
µ2 (0,1)
329
|I| (pA) 0
|I| (pA)
(2,-1)
VSD = +2 mV
(1,1)
478
8 c)
|I| (pA) 1
(1,1)
348
400
8 b)
(0,2)
540
FIG. 1. (a) False-color atomic-force microscopy image of the
device (left) and schematic depiction of the gate configuration
(right). (b) Current I vs. Vg3 and Vg5 with Vg2 = 2585 mV,
Vg4 = 2210 mV, Vg6 = 3195 mV, and VSD = 2 mV. White
dashed lines are guides to the eyes for the honeycomb edges.
Blue arrows represent EC , and the gaps between adjacent arrows indicate an additional Eorb . Circles mark triangle pairs
exhibiting PSB. (m, n) denotes the effective hole occupation
m and n on the left and right dot, respectively.
bottom row, but Eadd = 10.2 ± 0.1 meV in the middle
row. The increased Eadd in the right column and middle
row can be readily explained by the filling of a new orbital in the corresponding dot, so that in addition to the
(classical) charging energy EC the (quantum-mechanical)
orbital energy Eorb has to be taken into account, with
Eorb = 0.5 ± 0.1 mV in the left and Eorb = 0.7 ± 0.1 mV
in the right dot. This filling of a new orbital for both
dots allows us to identify the charge occupation (m, n)
with the occupation numbers of the newly filled orbitals
in the left and right dot for m, n ≥ 0.
For spin- 12 particles filling the orbitals, the bias triangle pairs marked by red circles in Fig. 1(b) should
exhibit PSB in opposite VSD directions. Figures 2(a)
and (b) display zooms of these triangle pairs for positive and negative VSD . The lower panels of Fig. 2(a) and
(b) display line cuts along the dotted lines for the VSD
direction with (blue) and without (green) PSB. A sup-
pression of the current at a detuning lower than the
singlet-triplet splitting ∆S-T can be observed at negative VSD in Fig. 2(a), where the zero-detuning current
at positive VSD of Ipos ( = 0) = 3.6 pA is reduced to
Ineg ( = 0) = −1.6 pA at negative VSD . Current suppression at positive VSD is visible in Fig. 2(b), where
Ipos ( = 0) = 1.1 pA and Ineg ( = 0) = −4.2 pA, exactly as expected. The current is thus only partially suppressed; comparable values for the resonant ( = 0) leakage current in double quantum dots have been reported in
the literature [22–24]. From the line traces taken in the
spin-blocked bias direction, we extract a singlet-triplet
splitting ∆S-T = 0.5 ± 0.1 meV. Note that this value is
very close to Eorb (see above). This is reasonable since
the (2,0) and (0,2) singlet-triplet splittings involve states
originating from successive quantum dot orbitals.
We retune the device by lowering Vg2 and Vg6 in
small, controlled steps and following the bias triangle
pair at the (1,1)–(2,0) degeneracy in Vg3 -Vg5 gate space
[Fig. 2(c)]. From the line cuts (lower panel) we extract
Ipos ( = 0) = 10.4 pA and Ineg ( = 0) = 1.2 pA, i. e., the
non-blocked current is almost threefold higher after retuning, whereas the leakage current even reduced so that
current rectification is now more pronounced.
In conclusion, Figs. 1 and 2 display the formation
of a gate-defined double quantum dot, in which we show
orbital shell filling for both dots and extract orbital energies of 500–700 µeV. We find PSB for two bias triangle
pairs in opposite bias directions.
3
a)
Magnetospectroscopy along Bz
~ parallel to the nanowire
We start our discussion with B
axis. The zero-detuning line cut [left panel in Fig. 3(b)]
has a maximum at B = 0 and decreases for increasing
magnetic field. Similar Ileak (B) curves with peak widths
of several hundred millitesla have been reported in other
systems [31–33] and were explained by spin-flip cotunneling [34]. We can fit the peaks to
Ico (B) = Ires +
4ecg ∗ µB B
,
∗
3 sinh g kµBBTB
(1)
with c = πh [{Γr /(∆−)}2 +{Γl /(∆+−2UM −2eVSD )}2 ],
where Ires is the residual leakage current, e the electron
charge, g ∗ the effective g factor, h Planck’s constant,
Γl,(r) the tunnel coupling with the left (right) reservoir,
∆ the depth of the two-hole level, and UM the mutual
charging energy [34]. The fit in the left panel of Fig. 3b
gives gz∗ = 0.4 ± 0.1 with a hole temperature T = 40 mK.
The finite residual current of Ires = 0.8 pA at high magnetic fields indicates a second leakage process efficient at
finite magnetic field, e. g. spin-orbit interaction [35]. The
By
Bx
-9
T0
0
I (pA)
ε0 (meV)
S
0
b)
ε=0
ε=0
ε=0
-2
-1
0
-150 150
0
-1
d)
Bz
aNW
E
0 Bz (T)1-1
c)
-0.5
Bx
By
0 By (T) 1-1
ε=150µeV
0 Bx (T) 1
ε=150µeV
I (pA)
Our quantum dots are elongated along the nanowire
~
axis ~aNW and are exposed to a static electric field E
from the bottom gates pointing out of the chip plane.
We explore the origin of the leakage current Ileak in PSB
by performing magnetospectroscopy measurements along
the white dashed line in Fig. 2(c) and plot Ileak versus
the detuning and the magnetic field in Fig. 3(a). In order to investigate possible anisotropic effects, we conduct
~ [see
measurements along three orthogonal directions of B
~
~
also Fig. 3(d)]: (Bz ) B parallel to the nanowire, (By ) B
perpendicular to both the nanowire and the electric field,
~ perpendicular to the nanowire and parallel
and (Bx ) B
to the electric field.
~ Here,
In Fig. 3(a) we plot Ileak vs 0 and B ≡ |B|.
0 ≡ (B = 0) is introduced as an absolute energy scale,
since is only defined relative to the alignment of the
(2,0) and (1,1) ground states. For all three Ileak (0 , B)
plots in Fig. 3(a), we scan the Ileak (0 ) line traces and set
the center of the lowest-energy peak as = 0. In Figs. 3(b)
and (c) we plot line traces for the leakage current at constant detuning as indicated in the panels. The green
dashed lines in Fig. 3(a) are guides to the eye for (B) = 0
(‘S-onset’). The lifting of the blockade at = ∆S-T is indicated by the blue dashed lines (‘T0 -onset’). The shift
of the S-onset and the T0 -onset to positive 0 -values for
increasing B in the plots has also been observed in other
experiments [25–27] and is explained by orbital effects
[25, 28, 29]. A more detailed discussion can be found in
the Supplemental Material S1 [30].
Bz
1
I (pA)
ANISOTROPIC LEAKAGE CURRENT
0
-1
0 By (T) 1-1
0 Bx (T) 1
FIG. 3. (a) Magnetospectroscopy measurements for different
magnetic field directions. Vg3 is swept from 486.0 to 489.0 mV
while keeping Vg5 = 360.0 mV fixed and sweeping B from 1
to -1 T. (b) and (c) line cuts from a) at fixed (open circles)
alongside fitted curves (solid lines). The inset shows a highresolution scan ofthe central I(Bx )-peak, Bx is here plotted
in mT. (d) Schematic depiction of the magnetospectroscopy
directions.
obtained g factors for a magnetic field applied parallel to
the nanowire axis are consistent with our findings for a
single quantum dot [15], where gz∗ = 0.2 ± 0.2.
~ pointing along the nanowire
To summarize, for B
axis we can explain the peak of Ileak at B = 0 with spinflip cotunneling and obtain gz∗ = 0.4(1), and we find hints
for additional leakage processes above ±0.8 T.
Magnetospectroscopy along By
We now let the magnetic field point perpendicular
to both the nanowire and the electric field. The zerodetuning line cut suggests the superposition of at least
two peaks: a narrow peak dominates up to B ≈ 0.2 T,
and is superimposed on a broader peak that is visible
over the whole range from –1 to 1 T and cannot be explained without further investigation. At finite detuning
[left panel in Fig. 3(c)] this broader peak is not observed,
which permits us to perform a more precise fit of the central peak. By fitting to Eq. (1) we obtain gy∗ = 1.2 ± 0.2
at = 150 µeV. The g factor along this magnetic field direction is significantly lower than in single quantum dot
measurements [15], where gy∗ = 2.7 ± 0.1 was maximal.
4
Because gy > gz , the spin-flip cotunneling induced
leakage current is exponentially suppressed for magnetic
fields above B ≈ 0.25 T and we obtain Ires at = 150 µeV
of ∼ −0.1 pA. At zero detuning, the minimum observed
current is ∼ −0.2 pA, which might be overestimated because of the additional features at high magnetic fields.
In summary, for B ⊥ E, ⊥ NW we observe a peak
in the leakage current at B = 0, which we explain with
spin-flip cotunneling and find gz∗ ≈ 1.2. The remaining
~ k ~aNW .
leakage current is significantly lower than for B
Magnetospectroscopy along Bx
The third high-symmetry direction is Bx , parallel to
E and perpendicular to the nanowire. Similar to the
other two magnetic field directions we find a peak around
B = 0 in the Ileak (B)-curve at = 0 [right panel of
Fig. 3(b)]. We fit Eq. (1) to a high-resolution magnetic
field sweep at a ten times lower sweeping rate (5 mT/min
instead of 50 mT/min) [inset of right panel in Fig. 3(b)],
which results in gx∗ = 3.9 ± 0.1. gx∗ here is substantially
higher than the gy∗ values obtained for B ⊥ E, as opposed to our findings in single quantum dots [15], where
gy∗ > gx∗ . One possible explanation for this discrepancy
is that it is very likely that we do not operate in the
lowest-energy subband of the nanowire. Here, we estimate approximately 70 holes to reside in each quantum dot. The theoretical calculations for the g factor
anisotropy [14] that we have confirmed experimentally
take into account only the quasi-degenerate two lowestlying subbands, and other theoretical work suggests that
the heavy-hole light-hole mixing is very different for different subbands [12, 36]. The measured angle dependence
of g ∗ confirms that the measured g ∗ -factor anisotropy is
not related to the crystal orientation but to the onedimensional confinement in the nanowire and a finite
electric field perpendicular to the nanowire axis.
The leakage current at magnetic fields B > 0.1 T,
where spin-flip cotunneling is efficiently suppressed, exhibits a qualitatively different behavior than for By : Up
to B ≈ 0.3 T the leakage current is minimal and increases
again for B > 0.3 T. Up to B = 1 T the leakage current
does not fully saturate, although the slope reduces at
both = 0 and = 150 µeV when B approaches 1 T,
which hints at a saturation for B > 1 T. Such an increasing Ileak (B) with saturation at higher B is again
an indication for spin-orbit induced leakage. Taking the
line trace at = 0 in the right panel of Fig. 3(b), we find
a minimal leakage current of Imin = 0.1 ± 0.1 pA. The
leakage current due to spin-orbit coupling between (2,0)
triplet and (1,1) singlet states can be expressed as
ISOI = Imax 1 −
2
8BC
2)
2
9(B + BC
,
(2)
where Imax is the maximum leakage current at high magnetic fields, and BC the width of the characteristic dip
around B = 0 [35]. If we now assume the minimum
leakage current Imin to be exclusively due to a spinorbit interaction we expect Imax = 9Imin = 1 ± 1 pA
at high fields. This estimate is in reasonably good
agreement with the value for Imax we obtain by fitting
the data excluding the central peak to Eq. (2) [see red
solid line in the right panel of Fig. 3(b)], where we find
Imax = −1.6±0.3 pA and BC = 1.0±0.2 T. Since at zero
detuning Imax = 4eΓrel , where Γrel is the spin relaxation
rate [35], we can calculate Γrel = 2.5 ± 0.5 MHz. This is
comparable with reports on measurements of heavy holes
in intrinsic Si [33], where Γrel = 3 MHz, and electrons in
InAs [24], where Γrel ranges from 0 to 5.7 MHz. We note
that BC is of the order of ∼ 1 T. Danon and Nazarov
[35] neglect the Coulomb interaction effects of B, which
are of relevance at such field strengths (see also the Supplemental Material S1). Therefore also other spin-orbit
or Coulomb related effects can provide significant leakage
paths [37–41].
To sum up, the spin-flip cotunneling peak in
Ileak (B, = 0) can be efficiently quenched by a magnetic
field due to a very high effective g factor of gx∗ ≈ 3.9. gx∗
is significantly higher than gy∗ , in contrast to our findings
in single quantum dots, where we see the opposite behavior. At higher B we notice an increasing Ineg , which we
assign to spin-orbit coupling induced mixing of the spin
states.
Let us now briefly compare our findings with existing literature and discuss the implications for spin-orbit
qubits. Pribiag et al. [42] measured the leakage current
at three different angles in plane of the sample and found
an almost isotropic dependence of the SOI-induced leakage current. To our knowledge there are no reports of
an angle dependence in the plane perpendicular to the
nanowire. Spin-flip cotunneling limited leakage current
as found in our device is exponentially suppressed by a
~ i. e. the
Zeeman splitting of the spin states at finite B,
~
remaining leakage current at a given B depends on the effective g factor. Since we find g ∗ to be highly anisotropic
~ and ~aNW , the leakage current can
with respect to both E
~ along E.
~ Also the leakage
be minimized by pointing B
current caused by SOI is anisotropic and depends on the
wave function overlap between the two dots [35]. Therefore it is possible to tune the double quantum dot so
that the SOI leakage current dip around B = 0 is wider
than the leakage current peak around B = 0 caused by
~ k E.
~
spin-flip cotunneling, which is the case here for B
Previously measured spin relaxation times of several hun~ along the nanowire [19] might
dred µs obtained with B
be extended by an order of magnitude when measured
~ according to the data presented here.
parallel to E
5
CONCLUSION
In conclusion, we have successfully formed a gatedefined double quantum dot in a Ge-Si core-shell
nanowire. We have observed shell filling and a Pauli spin
blockade, and have been able to explain the observed
leakage current by a combination of spin-flip cotunneling
at low magnetic fields and SOI-induced coupling between
singlet and triplet states at higher fields.
With these results we show that by wisely choosing
the magnitude and direction of a magnetic field applied to
a Pauli spin-blocked double quantum dot, one can achieve
longer spin lifetimes.
We thank Stevan Nadj-Perge, Sergey Amitonov, and
Paul-Christiaan Spruijtenburg for fruitful discussions.
We acknowledge technical support by Hans Mertens.
F.A.Z. acknowledges financial support through the EC
Seventh Framework Programme (FP7-ICT) initiative under Project SiAM No. 610637, and from the Foundation
for Fundamental Research on Matter (FOM), which is
part of the Netherlands Organization for Scientific Research (NWO). E.P.A.M.B. acknowledges financial support through the EC Seventh Framework Programme
(FP7-ICT) initiative under Project SiSpin No. 323841.
∗
Corresponding author, e-mail: m.brauns@utwente.nl
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Anisotropic Pauli spin blockade in hole quantum dots
Supplementary Material
Matthias Brauns,1, ∗ Joost Ridderbos,1 Ang Li,2 Erik P. A. M.
Bakkers,2, 3 Wilfred G. van der Wiel,1 and Floris A. Zwanenburg1
NanoElectronics Group, MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2
Department of Applied Physics, Eindhoven University of Technology,
Postbox 513, 5600 MB Eindhoven, The Netherlands
3
QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
(Dated: August 2, 2016)
Bz
∆S
0
∆S
T0
0T
0
By
T0
1T
ε0 (meV)
T0
Bx
∆S
T0
T0
1
0
1T
ε0 (meV) 1
∆T0
T0
0T
0T
0
Main text
Figure 2a
a)
ε0 (meV)
1T
1
FIG. 1. I(0 ) line cuts from Fig.3 in the main text at B = 0 T
(black curve) and B = 1 T (red) for all three magnetic field
directions. ∆S denotes the shift of the S-onset, ∆T0 the shift
of the T0 -onset.
S1 - ANISOTROPIC COULOMB INTERACTION
We start our discussion with the magnetospectroscopy
measurements for Bz and By (left and middle column in
Fig. 3a of the main text). In supplementary Fig. 1 we plot
I(0 ) line cuts at B = 0 T and B = 1 T. The T0 -onset
only changes minimally between 0 and 1 T, whereas the
S-onset shifts to positive 0 -values by ∆S = 100 ± 10 µeV
for Bz and by ∆S = 200±10 µeV for By . As stated in the
main text this behaviour has also been observed in other
experiments [3–5]. It can be explained by orbital effects
of the magnetic field [3, 6, 7], where B leads to a shrinking
of the wave function which increases the Coulomb energy
cost of singlet states as compared to triplet states and
therefore leads to a shift of the S-onset to higher energies.
A peculiarity of the measurements for Bx is the
T0 -onset, which, in contrast to the measurements along
By and Bz , shifts to higher 0 for increasing B. Between B = 0 and B = 1 T this shift amounts to
∆T0 = 280 ± 10 µeV, whereas the shift of the S-onset
sums up to ∆S = 330 ± 10 µeV. For Bz and By , we only
find ∆S = 100 ± 10 µeV and ∆S = 200 ± 10 µeV, respectively. This enhanced ∆S along with the pronounced
∆T0 for relatively small magnetic fields B ≤ 1 T indicate a change in the Coulomb interaction between the
two holes as compared to the measurements for By and
Bz [3, 8]. Theoretical calculations suggest an anisotropic
character of hole-hole interactions with respect to the
4
b)
Main text
Figure 2b
c)
Main text
Figure 2c
10
-4
I (pA)
5
I (pA)
arXiv:1608.00111v1 [cond-mat.mes-hall] 30 Jul 2016
1
0
0
0.5
ε (meV)
0
0
0.5
ε (meV)
0
0
0.5
ε (meV)
FIG. 2. Zooms of the I() line cuts for the unblocked bias
direction in the lower panels of Fig. 2 in the main text.
electric-field axis in Ge-Si core-shell nanowires [9], but
further studies are necessary for a definite conclusion on
the cause for the observed anisotropic B-dependence of
the two-hole spin states.
S2 - TUNNEL COUPLINGS
From the I()-curves in the non-blocked bias direction
(lower panels of Fig. 2 in the main text), we can extract
the tunnel coupling Γ to the reservoirs and the interdot
tunnel coupling t by fitting to the expression
2
2
I() = (Γ|t| )/((/h)2 + Γ2 /4 + 3|t| ) ,
(1)
where h is the Planck constant [1]. We fit the I()-curves
to Equation 1 only for ≤ 0, because for > 0 inelastic
processes that are not taken into account lead to an increased current, as can be inferred from the asymmetry
of I() around = 0 in supplementary Fig 2 where we
plot the fits alongside the data (the same approach has
been used e. g. by Li et al. [2]). The extracted values for
Γ and t are:
a) Γ = 100 ± 10 µeV, t = 1.6 ± 0.2 µeV
b) Γ = 90 ± 10 µeV, t = 1.6 ± 0.2 µeV
c) Γ = 100 ± 10 µeV, t = 2.7 ± 0.2 µeV
These values suggest that by changing the barrier voltages between the left and the right panel the interdot
2
tunnel coupling t increases significantly from 1.6 µeV to
2.7 µeV. This increase reflects an increased wavefunction
overlap between states in the left and the right dot, which
can be explained by the finite cross capacitance between
g2/g6 and the quantum dots which may affect the shape
of the wave functions. In all three cases t is more than an
order of magnitude smaller than Γ and thus the dominant
factor for the measured current.
Let us now take a more careful look at how the tunnel couplings relate to the leakage current in Pauli spin
blockade. Although t, which is the tunnel coupling limiting the current through the double quantum dot, is significantly higher in the right panel than in the left panel,
Ileak even decreases (see Fig. 2 in the main text). This
means that an increased overlap between (1,1) and (2,0)
states does not play a major role in the process that determines the leakage current.
∗
Corresponding author, e-mail: m.brauns@utwente.nl
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